Locating Objects in a Sensor Grid

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1 Locating Objects in a Sensor Grid Buddhadeb Sau 1 and Krishnendu Mukhopadhyaya 2 1 Department of Mathematics, Jadavpur University, Kolkata , India buddhadebsau@indiatimes.com 2 Advanced Computing and Microelectronics Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata , India krishnendu@isical.ac.in Abstract. Finding the location of an object, other than the sensor in a sensor network is an important problem. There is no good technique available in the literature to find the location of objects. We propose a technique to find the location of objects in a sensor grid. The locations of the sensors are assumed to be known. A sensor can only sense the number of objects present in its neighborhood. This small information injects low traffic in the network. The computations are carried out completely in the base station. 1 Introduction Micro-sensor is a small sized and low powered electronic device with limited computational and communicational capability. A Sensor Network [1] is a network containing some ten to millions of such micro-sensors (or simply sensors). Location finding is an important issue [4, 5, 6, 7] in a sensor network. It usually involves transmission of huge information and a lot of complex computations. Heavy transmission load drains the energy and shortens the life of the network [4]. An efficient location finding technique is required that involves little information passing, like only the counts, angles or distances of objects. Counting of objects needs simpler mechanism and hardware than those for angles or distances etc. Several techniques are available for finding the location of a sensor with unknown position [2, 8, 10, 11]. But finding the location of objects has not received much attention. We propose a technique for finding locations of objects based only on the counts of objects sensed by different sensors. We formulate a system of linear equations for this purpose. The variables in the system are binary in nature. Standard techniques for solving a system of linear equations takes polynomial time whenever the system has a unique solution. Otherwise, the system gives an infinite number of solutions. But in our case, usually the system of equations does not possess a unique solution. But the number of solutions should be finite, as the variables are binary in nature. One can determine the positions of A. Sen et al. (Eds.): IWDC 2004, LNCS 3326, pp , c Springer-Verlag Berlin Heidelberg 2004

2 Locating Objects in a Sensor Grid 527 the objects in exponential time on the number of cells in a grid. Our algorithm finds the locations of objects with much better complexity. Section 2 describes the model and problem. In Section 3 we propose the algorithm to find locations of objects. Section 4 deals with the analysis of the algorithm. In Section 5 simulation results are presented. Finally, we present our conclusion in Section 6. 2 The Model and Problem Statement We assume that the geographical area under consideration is divided into a two dimensional grid[3] with the following assumptions: The sensors are static and their positions are known. The objects to be sensed are indistinguishable to the sensors. The field is split into equal sized and small rectangular regions. Each of the smallest regions is termed as a cell. Each cell has unique spatial co-ordinate in two dimensions. A cell can contain at most one sensor and at most one object. A sensor can sense only 9 cells (the cell in which it resides and the eight neighboring cells). A sensor is capable of counting the number of objects in its neighborhood. A sensor node communicates its location and the count of the objects sensed, to the base station. The sensors may be placed in any manner, either by some design or through random placements. After placing the sensors, the positions of these sensors are determined with some localization technique. In this work, we assume that once the placement is complete, the sensors remain static. We consider a two dimensional field. The field is divided into m n identical cells. A cell can be uniquely identified by a pair of integers (i, j) when the cell lies in ith row and jth column in the field. We assume that a sensor or an object of our interest is small enough to fit inside a cell. A sensor detects an object by optical, ultrasound or radio energy reflection or by any other mean. Intensity of energy decreases with the increase in distance. Therefore, a sensor can not sense objects placed beyond a finite region around it. This region is termed as the locality or sensing region of the sensor. We assume that the locality consists of 9 cells. The locality of a sensor at (u, v) consists of the cells (u, v), (u, v ± 1), (u ± 1,v) and (u ± 1,v± 1). We introduce a binary variable x uv corresponding to the cell (u, v). The value of 1 is assigned to a variable, x uv, if and only if the cell (u, v) contains an object and value 0 for no object. x uv indicates the number of objects in the cell (u, v). If c is the count of the objects recorded by the sensor at (u, v) then, x u 1 v 1 + x u 1 v + x u 1 v+1 + x uv 1 +x uv + x uv+1 + x u+1 v 1 + x u+1 v + x u+1 v+1 = c (1)

3 528 B. Sau and K. Mukhopadhyaya The equation (1) contains 9 variables. This equation alone is not enough for finding the location of objects. A sensor sends own position and the count of objects in its locality to the base station. The base station generates many equations with information from different sensors. Even then, the base station may not find the exact locations of all objects. Our proposed algorithm calculates the probabilities, p uv, that the cell (u, v) contains an object. The problem to the base station may be stated formally as: Problem 1. Suppose, the field is equipped with p sensors, s 1, s 2,, s p placed in a two dimensional grid field of size m n. Let (u i,v i ) be the location of the sensor s i in the grid. c i denotes the number of objects observed by s i. Note that a sensor at the border of the grid does not have 9 neighbors. Using equation (1) we obtain a system of linear equations as: x ui 1 v i 1 + x ui 1 v i + x ui 1 v i+1 + x ui v i 1 +x ui v i + x ui v i+1 + x ui+1 v i 1 + x ui+1 v i + x ui+1 v i+1 = c i (2) for i =1, 2,...,p where x uv {0, 1} and x uv = 0 for the cell (u, v) beyond the grid. Find the complete set of probabilities, p uv, u =1, 2, m, v =1, 2, n. Fig. 1 shows a scenario of a 4 5 sensor grid. A bullet in a cell indicates that the cell contains an object. The grid contains 6 sensors s 1, s 3, s 4, s 5 and s 6 at locations (1, 2), (2, 3), (2, 4), (3, 4), (4, 2) and (4, 5). They found 0, 3, 2, 2, 2 and 0 objects respectively s 5 s 6 s 4 s 2 s 3 s Fig. 1. An example showing sensors marked by s i s and objects denoted by The system of equations corresponding to the scenario in Fig. 1 is: x 11 + x 12 + x 13 + x 21 + x 22 + x 23 =0 x 12 + x 13 + x 14 + x 22 + x 23 + x 24 + x 32 + x 33 + x 34 =3 x 13 + x 14 + x 15 + x 23 + x 24 + x 25 + x 33 + x 34 + x 35 =2 x 23 + x 24 + x 25 + x 33 + x 34 + x 35 + x 43 + x 44 + x 45 =2 x 31 + x 32 + x 33 + x 41 + x 42 + x 43 =2 x 34 + x 35 + x 44 + x 45 =0 The system of equations will be solved to find the probabilities, p uv.

4 Locating Objects in a Sensor Grid The Algorithm While solving the system of equations (2) we take the advantage of the binary nature of the variables. The set of all possible solutions to the system is termed as the solution set and denoted by S. Three markers A, B and C are used against each variable, x uv. A indicates x uv takes any value irrespective of the values of other variables in the solution set. B denotes x uv may take 0 or 1 depending on the values of other variables in the solution set. C indicates that x uv may take only one of 0 and 1 whatever may be the value of the other variables in the solution set. Our algorithm counts the number of solutions rather than finding the solution set to the system. Algorithm 1 (ObjectLocation:) Step-1: Initially, each variable is marked A and S =. Step-2: For i =1to p perform Step-3 through Step-5. Step-3: Let S i be the set of the variables in the in the i-th equation. Let B i be the set of those variables in S i which are marked B. Step-4: For each of solution s S do From the solution s, take out the portion corresponding to the variables in B i, say s. If (the number of 1 s in this portion exceeds c i ) Remove s from S. Else /* this portion satisfies the i-th equation */ Find all possible combinations of 0s and 1s for the variables in S i satisfying the i-th equation. Insert the combinations into S by replicating the combination for the variables marked B. End if Step-5: In the i-th equation the variables marked A are changed to B. If any variable marked B has only one value (0 or 1), use the mark C for it and rewrite the system accordingly. Step-6: Let b be the size of the solution set. For each cell (u, v) { xuv when x Compute : P uv = uv is marked C when x uv is marked B Step-7: End. c uv b /* Note that no variable remains marked A at the end of computation as each variable is seen by at least one sensor. */ 4 Analysis of the Algorithm The first step in the algorithm take O(mn) time. Steps 2, 3 and 5 take O(1) time. Step 4 is repeated for each solution. Therefore, Step 4 has the time complexity of

5 530 B. Sau and K. Mukhopadhyaya O(the number of solutions in the solution set). Step 2 through Step 5 will be repeated p times (where p is the number of sensors). Step 6 takes another O(mn) time. The time complexity of the algorithm is O(mn + p {the number of solutions in the solution set}). If p = O(mn) the time complexity of the algorithm becomes O(mn {the number of solutions in the solution set}). Traditional methods solve the system in O(mnp) time when the number of solution is unique. In case of unique solution, the time complexity of our algorithm becomes O(mn). 5 Simulation Results We simulated our algorithm on a computer and counted the number of operations. The results are shown in Tables 1. For convenience, we use the following symbols in the tables: m, n and p denote the numbers of rows, columns and sensors in the grid field respectively. For an m n grid, AvgSol and MaxSol denote the expected and maximum number of solutions to the system. OpCount and M axop indicate the expected and maximum number of operations (comparison operations to check the validity as mentioned in the step 4 in the algorithm) to solve the system. We note that the simulation results are consistent with our analysis. Though there can be a large number of operations in the worst case (MaxOp), the average operation count (OpCount) is far lower than that. The number of operations increases as the density of sensors goes down. That can be attributed to the fact that lower sensor density leads to higher number of cells on which we cannot make a definite conclusion. m n p AvgSol mn AvgSol Table 1. Results for p = 70% of mn mnp AvgSol OpCount MaxSol mn MaxSol mnp MaxSol MaxOp

6 Locating Objects in a Sensor Grid Conclusion The technique, proposed in this paper, considers only the location of a sensor the number of objects in its locality. The required hardware for this would be far simpler than those for distance or angle estimation. The proposed algorithm takes time of the order of the number of cells in the grid times the number of different solutions to the system. A careful selection of data structure can achieve better time complexity. A sensor may consult with neighboring nodes to reduce the data for communication over the network. Our Algorithm is devised for a Sensor Grid and in future we plan to devise techniques for object location in an arbitrary sensor field. One can also address the problem of object location when the cells can contain multiple objects or the sensors or objects are mobile. References 1. Akyldiz, I.F., Su, W., Sankarasubramaniam, Y., Cayh ci, E.: Wh eless Sensor Networks: A Survey. Computer Networks, Vol. 38, No. 4. March (2002) Bulusu, N., Heidemann, J., Estrin, D.: GPS-less Low Cost Outdoor Localization For Very Small Devices. IEEE Personal Communications Magazine, Vol. 7, No. 5. October (2000) Chakrabarty, K.,Iyenger, S.S., Qi, H., Cho, E.: Grid Coverage for Surveillance and Target Location in Distributed Sensor Networks. IEEE Transactions on Computers, vol. 51, No. 12. October (2002) Dasgupta, K., Kalpakis, K., and Namjoshi, P.: Improving the Lifetime of Sensor Networks via Intelligent Selection of Data Aggregation Trees. Proc. of the SCS Comm. Networks and Distributed Systems Modelling and Simulation Conf. (CNDS 03). Orlando, Florida January (2003) 19 23, 5. Heinzelman, W., Chandrakasan, A., and Balakrishnan, H.: Energy-Efficient Communication Protocol for Wireless Microsensor Networks. Proc. of the Hawaii International Conference on Systems Sciences. Maui, Hawaii January (2000) Li, X.Y., Wan, P.J., Frieder, O.: Coverage in Wireless Ad-hoc Sensor Networks. ICC (2002) 7. Meguerdichian, S., Koushanfar, F., Qu, G., Potkonjak, M.: Exposure In Wireless Ad-Hoc Sensor Networks. Proc. 7th Int. Conf. on Mobile Computing and Networking (MobiCom 01). Rome, Italy July (2001) Priyantha, N., Chakraborty, A., Balakrishnan, H.: The Cricket Location-Support System. Int. Proc. of ACM MobiCom. ACM, Boston, MA (2000) Raghunathan, V., Schurgers, C., Park, S., Srivastava, M.B.: Energy-aware wireless microsensor networks. IEEE Signal Processing Magazine, vol. 19, No. 2. IEEE March (2002) Savvides, A., Han, C. M., Srivastava, B.: Dynamic Fine-Grained Localization in Ah-Hoc Networks of Sensors. Int. Proc. of ACM MOBICOM. ACM, Rome, Italy (2001) 11. Thrun, S., Fox, D., Burgard, W., Dellaert, F.: Robust Monte Carlo Localization for Mobile Robots. Artificial Intelligence (2001)